Preheat the oven to 175°C.
Combine the butter and graham crackers in the bowl of a food processor and pulse until finely ground and the mixture holds together when pinched. Transfer to a 9-inch tart pan with a removable bottom, and press evenly along the bottom and about halfway up the sides. Place on a baking sheet and bake until lightly golden, about 10 minutes. Let cool completely.
Combine 1/3 cup of the sugar, the sour cream and cream cheese in a stand mixer fitted with a paddle attachment (or a large bowl if using a hand mixer), beat until smooth. Add the lemon zest, vanilla and eggs, and continue to beat until incorporated. Pour into the cooled tart shell and bake on a baking sheet until set, 25 to 30 minutes. Cool to room temperature and refrigerate until chilled, at least 1 hour up to overnight.
About 30 minutes before serving, toss the berries with the remaining 2 tablespoons of sugar, mash slightly with a wooden spoon and let sit at room temperature. Spoon the berries and collected juice over the tart. Serve.